Forced oscillations of nonlinear damped equation of suspended string

Abstract

We shall study the existence of time-periodic solutions of nonlinear damped equation of suspended string to which a periodic nonlinear force works. We shall be conterned with weak, strong and classical time-periodic solutions and also the regularity of the solutions. To formulate our results, we shall take suitable weighted Sobolev-type spaces introduced by [M. Yamaguchi, Almost periodic oscillations of suspended string under quasiperiodic linear force, J. Math. Anal. Appl. 303 (2) (2005) 643-660; M. Yamaguchi, Infinitely many time-periodic solutions of nonlinear equation of suspended string, Funkcial. Ekvac., in press]. We shall study properties of the function spaces and show inequalities on the function spaces. To show our results we shall apply the Schauder fixed point theorem and the fixed point continuation theorem in the function spaces.

Global smooth solutions of IBVP to nonlinear equation of suspended string

“…This is the Poincaré type inequality in H 1 0 (0, a; x m ). This is a corollary of Proposition 2.3 in [7].…”

“…(The definitions will be given in this section later. See also [6,7].) In this paper we shall take m = 0 that corresponds to the uniform density.…”

“…Then combining the energy estimates of the higher order timederivatives of approximate solutions with the compactness arguments, we shall show that the approximate solution converges to a function that is a solution of (P) with higher order t-derivatives. The smoothness of the solutions shall be obtained by the above t-regularity of solutions and the x-regularity by the use of an elliptic technique shown in [8].…”

Global Classical Solutions of IBVP to Nonlinear Equation of a Suspended String

“…The operator L is a particular form of suspended string equations introduced by [K-G-S] and [Ko]. As for IBVP to the linear equation of the suspended string with the quasi-periodic forcing term, the existence of almost periodic C 2 -solutions was proved in [Ya1], and for several periodic problems of nonlinear equation of suspended string, see [Ya2] and [Ya-Na-Ma].…”

Global Solutions of IBVP to Nonlinear Equation of Suspended String

“…For the existence of infinitely many time-periodic solutions of nonlinear equations, see [Ya2]. See also [Ya-Na-Ma] for the periodic problem to nonlinear damped equations with periodic forcing term.…”